[/quote]
,
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
such that for any integer 
there exists an integer 
for which 
is divisible by 
are distinct prime numbers which satisfy
for natural number 
,
,
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
Y by Adventure10
We are given a right-angled triangle 
with right angle in 
. Let 
be the circle with center 
and radius 
, and let 
be the circle with center 
and radius 
. Let 
and 
be the common points of and the line 
, and let 
and 
be the common points of and the line with with between and . Prove that the line 
bisects the angle 
.
with right angle in 
. Let 
be the circle with center 
and radius 
, and let 
be the circle with center 
and radius 
. Let 
and 
be the common points of and the line 
, and let 
and 
be the common points of and the line with with between and . Prove that the line 
bisects the angle 
.This post has been edited 1 time. Last edited by parmenides51, Feb 2, 2020, 7:55 PM
Reason: wording corrected
Reason: wording corrected
are orthogonal. Hence, 
(By Self-Orthogonality Lemma) and as 
.
